A132774 A natural number operator.

1, 2, 3, 0, 4, 5, 0, 0, 6, 7, 0, 0, 0, 8, 9, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 14, 15, 0, 0, 0, 0, 0, 0, 0, 16, 17, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 23

Offset: 1

Author

Gary W. Adamson, Aug 28 2007

Keywords

• nonn
• tabl

Comments

Row sums = A016813: (1, 5, 9, 13, ...).

A132774 * [1, 2, 3, ...] = A033951.

Examples

			First few rows of the triangle are:
  1;
  2,  3;
  0,  4,  5;
  0,  0,  6,  7;
  0,  0,  0,  8,  9;
  0,  0,  0,  0, 10, 11;
  ...
		

Links

• Stefano Spezia, First 150 rows of the triangle, flattened

Crossrefs

Cf. A016813 (row sums), A033951, A060747 (main diagonal).

Programs

Mathematica

T[n_,k_]:=If[n==k,2n-1,If[n-k==1,2(n-1),0]]; Flatten[Table[T[n,k],{n,12},{k,n}]] (* Stefano Spezia, Dec 21 2021 *)
Join[{1},Flatten[{#,PadRight[{},#[[1]]/2,0]}&/@Partition[Range[2,30],2]]] (* Harvey P. Dale, Mar 24 2024 *)
Join[{1},Flatten[Table[Join[Range[2n,2n+1],PadRight[{},n,0]],{n,20}]]] (* Harvey P. Dale, Mar 25 2024 *)

Formula

As an infinite lower triangular matrix, (1, 3, 5, ...) in the main diagonal and (2, 4, 6, ...) in the subdiagonal; with the rest zeros.

From Stefano Spezia, Dec 21 2021: (Start)

T(n, k) = 2*n - 1 if n = k, T(n, k) = 2*(n - 1) if n - k = 1, otherwise T(n, k) = 0.

G.f.: x*y*(1 + x*(2 + y))/(1 - x*y)^2. (End)

Extensions

More terms from Stefano Spezia, Dec 21 2021